p-adic variation of L-functions of exponential sums, I

Abstract

For a polynomial $f(x)$ in $({\mathbb{Z}}_p\cap \mathbb{Q})[x]$ of degree $d>2$ let $L(f\mathrm{mod}p;T)$ be the $L$-function of the exponential sum of $f\mathrm{mod}p$. Let $\mathrm{NP}(f\mathrm{mod}p)$ denote the Newton polygon of $L(f\mathrm{mod}p;T)$. Let $\mathrm{HP}(f)$ denote the Hodge polygon of $f$, which is the lower convex hull in the real plane of the points $(n,n(n+1)/(2d))$ for $0\le n\le d-1$. We prove that there is a Zariski dense subset $\mathcal{U}$ defined over $\mathbb{Q}$ in the space ${\mathbb{A}}^d$ of degree-$d$ monic polynomials over $\mathbb{Q}$ such that for all $f$ in $\mathcal{U}(\mathbb{Q})$ we have ${\lim }_{p\to \infty }\mathrm{NP}(f\mathrm{mod}p)=\mathrm{HP}(f)$. Moreover, we determine the $p$-adic valuation of every coefficient of $L(f\mathrm{mod}p;T)$ for $p$ large enough and $f$ in $\mathcal{U}(\mathbb{Q})$, and that of $L(x^d+ax\mathrm{mod}p;T)$ for all $a\ne 0$.

This is the closest (and final) LaTeX version to the published version in American Journal of Mathematics (2003). Among many differences from v1, the published Section 6 under the title "Generic Newton Polygon for x^d+ax" gives its p-adic Newton polygon explicitly

References in corpus (1)

• An introduction to the theory of Newton polygons for L-functions of exponential sums

Cited by in corpus (1)

• Generic exponential sums associated to Laurent polynomials in one variable